Subgroups — C₃×#3756 #88847

Order 432.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#634	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#3756	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#216	—	3	4	no	maximal
6	C₃³×C₄	—	2	1	no	—
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃×C₃⋊Dic₃	—	2	1	no	—
9	C₃²×Q₈	—	2	4	no	—
10	C₃²×Q₈	D₃	1	4	yes	—
11	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
12	#634	C₂×C₃	1	1	yes	characteristic
13	C₃×Dic₆	—	3	4	no	—
14	C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
15	C₃×C₃⋊C₈	—	3	4	no	—
16	C₂×C₃³	D₄	1	1	yes	characteristic
17	C₃×Q₁₆	—	9	1	no	—
18	#216	—	3	4	no	—
19	C₃²×C₄	—	2	8	no	—
20	C₃²×C₄	—	2	4	no	—
21	C₃²×C₄	—	2	1	no	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	C₂×D₃	1	4	yes	—
24	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
25	C₃⋊Dic₃	—	2	1	no	—
26	C₃×Dic₃	—	6	4	no	—
27	C₃³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	2	4	no	—
29	C₃×Q₈	—	9	1	no	—
30	C₃×Q₈	C₃×D₃	1	4	yes	—
31	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
32	Dic₆	—	3	4	no	—
33	C₃×C₈	—	9	1	no	—
34	C₃⋊C₈	—	3	4	no	—
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	#67	1	4	yes	—
37	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
38	Q₁₆	—	9	1	no	Sylow
39	C₃×C₄	—	2	8	no	—
40	C₃×C₄	—	2	4	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	2	4	no	—
43	C₃×C₄	—	18	1	no	—
44	C₃×C₄	C₂×C₃×D₃	1	4	yes	—
45	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
46	Dic₃	—	6	4	no	—
47	C₃²	—	2	4	no	—
48	C₃²	#216	1	4	yes	complemented
49	C₃²	C₃×Q₁₆	1	1	yes	characteristic, complemented
50	Q₈	—	9	1	no	—
51	Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
52	C₈	—	9	1	no	—
53	C₂×C₃	—	2	4	no	—
54	C₂×C₃	C₃×#67	1	4	yes	—
55	C₂×C₃	#638	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	18	1	no	—
58	C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
59	C₃	—	2	4	no	—
60	C₃	C₃×#216	1	4	yes	complemented
61	C₃	#3756	1	1	yes	characteristic, direct product factor
62	C₂	C₃×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#638
commutator, cosocle	C₃²×C₄	C₂²×C₃
Frattini	C₄	C₂×C₃×C₃⋊D₃
Fitting	C₃³×Q₈	C₂
socle, prime radical	C₂×C₃³	D₄
hypercenter	C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₃×Q₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #3756	C₃	direct (N × H)
27. C₃³	Q₁₆	semidirect
48. C₃²	#216	semidirect
49. C₃²	C₃×Q₁₆	semidirect
60. C₃	C₃×#216	semidirect
61. C₃	#3756	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃³×Q₈	2
C₃×#634	2
C₃×C₃⋊(C₃⋊C₈)	2
#3756	3
C₃×#216	3

Subgroups — C3×#3756 #88847

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3756 #88847